To determine the densities of the metal and the unknown liquid, the weights of the metal in air, water, and the unknown liquid are used to apply the principle of buoyancy. The metal weighs 50 N in air, 36 N in water, and 41 N in the unknown liquid, allowing for calculations based on the buoyant force experienced in each medium. The density of the metal can be calculated using the formula d = m/v, where the mass is derived from the weight and gravitational acceleration. The discussion highlights that temperature and pressure can affect density, particularly for liquids, but the relevance of temperature changes in this specific problem is debated. Ultimately, accurate density calculations require careful consideration of buoyancy and the properties of the substances involved.